This book, which I offer to inquisitive readers, deals with the algorithm-theory problem of enumeration considered in discrete mathematics.
This book is part of a series that describes an artificial-mind model I developed based on the philosophy of the Kantian apriorism. This model imitates the human mind and human thinking.
The question of whether the complexity classes of P and NP problems are equal, also known as the enumeration problem, has been the central open problem in algorithm theory for more than four decades.
An affirmative answer to that question would mean it is theoretically possible to solve many complex problems far faster than they can be solved today.
The lack of a solution to the central problem of modern discrete mathematics inhibits the development of not only discrete mathematics but of mathematics as a whole.
This applies to many areas of study of mathematical structures in algebra, topology, geometry, and suchlike fields.
As with other enumeration problems, a feature of the traveling salesperson problem (TSP) is that it is fairly easy to obtain its solution. The hardest part is proving the optimal solution (finding the best solution).
This book considers a seamless method for exactly solving combinatorial-optimization problems (I call it an effective combinatorial-optimization method). I developed the method using, as an example, the TSP, an NP-hard problem in combinatorial optimization.
Exact, effective methods for solving NP problems are yet unknown.
In applying my effective combinatorial-optimization method, I was able to find an indirect indicator that the resulting solution is optimal.
It has been proved that a method that allows you to effectively solve an NP problem can also be applied to solving other NP problems.
My research yielded a certain relationship for the TSP. This relationship is natural, objective, and scientifically novel.
My effective combinatorial-optimization method is based on this relationship and uses the indirect criterion I found for determining the best solution. The method is also applicable to solving a variety of other NP problems.
I claim I might have solved the central problem of modern discrete mathematics — the P versus NP problem. P = NP.
This book opens a series devoted to developing an anthropomorphic artificial mind as a new line of research different from artificial intelligence (AI), whether strong or weak. This series of my books offers fundamentally novel insights and is the only such series in the scientific world.
The Artificial Mind
Once created, an artificial mind similar to the human mind will give humanity unlimited possibilities.
The artificial mind will have a different energy source. This energy source will not compete with the food chain of various biological beings, including humans.
The artificial mind will be able to live in those environments where humans cannot. It won’t need air to breathe. It will be able to live at almost any ambient pressure (from zero to hundreds of atmospheres) and will be easier to protect from heavy radiation.
The artificial mind will directly interact with computer and data-transfer systems.
The artificial mind can have sensory organs that humans don’t have. The functional spectra of these organs (hearing, seeing in the infrared, ultraviolet, and X-ray spectra; smell and tactile sensations) could be far wider than those in humans.
The artificial mind could have a body that differs from the human body in size, shape, and number and shape of limbs (manipulators).
The artificial body won’t age (it will be replaceable).
The artificial mind can be preserved and restored when lost, meaning the practical immortality of the mind (the individual).
The artificial mind can fully exist in the virtual world (a matrix).
The artificial mind can travel through space at the speed of light (between transmitter and receiver).
The artificial mind can function in a different timeline (e.g., one accelerated several times or, conversely, slowed down many times).
These possibilities will allow a manifold increase in the number of intelligent creatures in Russia without leading to overpopulation, and that in turn will increase Russia’s intellectual potential and accelerate scientific progress across the country.
Russia would be able to actively explore space and populate new planets, making distant space expeditions, terraforming planets to make them habitable for Earth life.
The artificial mind would help raise the standard of living in Russia, and improve the efficiency of human life, making for better use of natural resources.
My term artificial mind is chiefly based on solving the P versus NP problem.
Today the term artificial mind is understood as AI, a kind of intelligent system or program that has limited intellectual functions and can solve a narrow range of problems — that is, has a limited scope of application.
I believe there are fundamental differences between this definition of AI and that of the artificial mind.
Modern strong AI is, basically, a software program capable of thinking and being aware of itself — of itself as a separate entity — and in particular, of understanding its thoughts as a human would.
By contrast, the artificial mind is, in my opinion, not a program. It is a mechanism — mostly analogue one. A primary role in its development should be played by engineers in various fields of science and technology, not by programmers. The central role should go to system technology engineers and developers of various analogue and digital devices.
Programmers’ role is secondary, and it’s mainly of an applied nature, finding use in physical implementation of the artificial mind’s various functions.
I think the contemporary line of research called AI is a dead end. AI achievements can, at best, be applicable to the artificial mind only as a complement.
My Kantian term pure artificial mind denotes the absolute mind, or theoretical reason according to Kant, that is had not only by people but by other intelligent creatures, both on the Earth according to Kant and in the universe.
I believe, as Kant did, that we cannot assert that rationality (as an abstract metaphysical definition of the phenomenon of the human mind) can exist only in the human body on Earth.
If you ask whether the human being (a rational being) could appear in a different environment, in a different body, with different physical laws, perhaps in a space with a different number of dimensions, with a whole lot of other different conditions, then we have no reason to answer that question negatively.
You can suppose the manifestations of intelligence in that other creature would be in many ways different from those in the human being.
In my opinion, the phenomenon of rationality (as a kind of abstract, indefinite, yet metaphysically intuitive one) can exist in a wide possible range in the universe; therefore, it can be considered an ideal phenomenon according to Kant — pure reason.
It’s not autonomous in its full understanding (according to Kant, it needs a body). But the body and its properties are only the reason for the nature of how intelligence is manifested, which makes it practical according to Kant but does not determine its very existence.
A manifestation of intelligence is impossible without the existence of a body with some characteristics. To put it differently, there are no acceptable or unacceptable sets of body characteristics for the existence (manifestation) of intelligence.
With this in mind, we can observe manifestations of the mind only in humans and in the reality that we have.
But it should be said that one cannot narrow the phenomenon of rationality to these boundaries, trying to create an artificial mind by imitating individual manifestations of the human mind in the form of algorithms.
In other realities in the universe, intelligence may be manifested differently. There can be countless such variations of the mind.
As I see it, in creating algorithms for AI, both weak and strong, that imitate individual properties of the human mind and in combining them together, you cannot create an artificial mind.
Although we can isolate areas of the human brain and observe the various functions of those areas, human thinking doesn’t lend itself to decomposition.
There can be no algorithm for AI, since an algorithm solves a specific problem and the algorithm’s scope of application is confined to that problem.
I argue, according to Kant, that the artificial mind should be able to solve any problems — even those that didn’t exist when it was born. The algorithmic approach doesn’t make that possible, since a problem must be posed before its algorithm is written.
If we talk in a similar vein about neural networks, algorithms only emulate a neural network that is based on a different (nonalgorithmic) concept.
The creation of an emulation algorithm for a neural network doesn’t determine what tasks this neural network will solve. These problems will be determined by data obtained as part of teaching the network, and the structure of the neural network will affect its efficiency for certain categories of tasks and data.
Moreover, various configurations of a neural network are studied empirically, suggesting that at the time of designing the neural network, its effectiveness for solving a specific problem cannot be determined.
In one of his interviews, the American scientist Marvin Minsky, a founding father of the field of AI, drew parallels between the operating principle of the brain and the computer. He said it is impossible to understand what a computer is and how it works, even if you scrutinize the operating principle of the transistors that make up the computer. Similarly, you cannot understand how the brain works by studying the functioning of neurons, dendrites, and axons.
Thinking, or rationality, according to Kant, is the ability to exercise one’s will with regard to the probable outcome (result, consequence) of an event, based on subjective experience and with the aim of expanding the possibilities of one’s influence on them in the future.
AI, whether strong or weak, does not and cannot contain any prescriptions for the ability to judge, according to Kant. Indeed, since AI is abstracted from any contents of cognition, then its purpose comes down to the task of analytically explaining only one form of cognition in concepts, judgments, and inferences, thereby establishing formal rules for any application it might have.
The ability to judge and instruct, according to Kant, is a distinguishing feature of the human mind that cannot be reproduced by any AI. We could hammer into AI software any number of rules, but the ability to use them correctly is not inherent in it.
AI doesn’t have that ability. No rules dictated to AI from the outside for the purpose of judgement can guarantee AI won’t apply them erroneously.
Thus, while AI can know so many good medical, legal, technical, military, political, and other rules that it is capable of being a good teacher in those fields, it can easily make mistakes in applying the rules, because it lacks the natural human ability to judge.
So, although AI is able to perceive the general in abstracto, it cannot distinguish its fitness for a given scenario in concrete. The only, and enormous, benefit of rules and examples AI learns from the outside is precisely that they enhance AI.
Obtuse, narrow-minded AI lacking reason in developing its own concepts, can even achieve, through learning, some “erudition” in some areas of knowledge, but this erudition doesn’t save it mistakes (including, perhaps, quite ghastly ones).
My books (of which more will be said hereafter) present, in an easy-to-follow, clear, and at the same time precise and systematic manner, the basic concepts and activities of an anthropomorphic artificial mind and its machine will.
I argue that the anthropomorphic artificial mind I propose has the ability to judge and instruct according to Kant.
I have developed methods for the functioning of the anthropomorphic artificial mind and a strategy for creating this kind of mind.
I place special emphasis on the general idea of machine-will operations based on a practical rule I developed (the practical principle of apperception, the practical principle of anticipation, and a method for solving combinatorial problems as a way of understanding the real world by the practical artificial mind).
The logic of the practical artificial mind is shown in keeping with some of the attributes of the human mind and human thinking: the theory of the anthropology of reason and thinking by the renowned philosopher Immanuel Kant. The theory is described in his books:
— Critique of Pure Reason
— Groundwork of the Metaphysic of Morals
— Critique of Judgment
— Critique of Practical Reason
I have determined the composition of the machine brain according to Kant, which, along with reason, thinking, and intellect according to Kant, also includes intelligence as seen from the modern perspective on AI.
I have also described the surrounding world for the metamind, which, according to Kant, includes the outer world, the real world, the inner world of the metamind, and the soul of the metamind.
I have attempted to describe the definition and composition of the soul of the metamind, proving its simple nature as opposed to the soul of the human mind. From the perspective of Kant’s philosophy, my attempt to expand human knowledge goes beyond all possible experience. This, in his opinion, is impossible.
Kant was the first to point out the possibility of the existence on Earth of intelligent beings other than humans. Because of his free thinking, Kant’s contemporaries often criticized him and even laughed at him, failing to understanding what he meant by other rational beings.
For this reason or some other, Kant never debated with his contemporaries. Nor did he want to have disciples.
In his works, Kant addressed subsequent generations with the idea that they should pay close attention to his new science — metaphysics of nature — based on philosophical apriorism, in a departure from the philosophy propounded by the great Isaac Newton, who represented space as a mathematical abstraction in the form of a Cartesian system filled with ether.
Kant put special emphasis on his transcendental logic. His views on the topic are still relevant today.
The books I recommend that you read will give you a general understanding of the structure, concepts, and operating principle of the artificial mind.
The books will benefit smart, inquisitive, and well-disposed readers interested in the anthropomorphic artificial mind that uses AI simply as a complement.
This book series presents examples of solving various enumeration problems; the model of a specialized parallel hybrid computing machine; a model of intelligent behavior; a model of an anthropomorphic machine platform; a model of the metamind, its origin, foundations, and principles; and the practical artificial mind per se.
The computational process is presented, basically, as an analogue computational model, which allows solutions to be obtained almost instantly.
The Traveling Salesperson Problem
Combinatorial optimization (CO) consists in finding the optimal solution in a finite set of solutions — that is, combinatorial problems can be solved by exhaustive enumeration. For many CO problems, exhaustive enumeration is not possible.
At present, developing accurate, effective methods for intractable CO problems is considered problematic and unlikely. These problems include the TSP.
The problem is proved to be intractable and relates to NP-hard problems. For this class of problems, the existing enumeration problem in algorithm theory is open. It is argued that if this problem is solved, then the problems of the NP class can be solved using a single effective method, since these problems are reducible to each other. The existing exact methods for solving CO problems are, essentially, implicit-enumeration methods with exponential time complexity.
These methods include:
— The branch-and-bound method
— The dynamic-programming method
The methods are ineffective by definition because they require, for purposes of solving CO problems, a solution time that exponentially depends on the problem size. For that reason, approximate effective methods have been developed to solve CO problems. These include:
— Approximate algorithms with guaranteed quality estimates for the resulting solution
— Probabilistic algorithms
This book offers a nonenumeration method for exactly solving CO problems. Instead of enumerating, this method searches for the optimal solution based on the regularity inherent in CO problems and on the indirect indicator of the solution’s optimality.
The Idea of the Method
The TSP has factorial time complexity. The number of arrangements A in graph V is given by
Consider the ordered vectors of arrangements A in graph V.